**Proof.**
Let $\mathfrak q$ be a prime of $S$ lying over a prime $\mathfrak p$ of $R$. Assume that $\dim (S_{\mathfrak q}) \leq k$. Since $\dim (S_{\mathfrak q}) = \dim (R_{\mathfrak p}) + \dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q})$ by Lemma 10.112.7 we see that $\dim (R_{\mathfrak p}) \leq k$ and $\dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) \leq k$. Hence $R_{\mathfrak p}$ and $S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}$ are regular by assumption. It follows that $S_{\mathfrak q}$ is regular by Lemma 10.112.8.
$\square$

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